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$$\partial$$-reducing Dehn surgeries and 1-bridge knots. (English) Zbl 0788.57005
Suppose $$K$$ is a knot in a $$\partial$$-reducible 3-manifold $$M$$ such that no compressing disk of $$\partial M$$ is disjoint from $$K$$. A Dehn surgery on $$K$$ is called $$\partial$$-reducing if the surgered manifold is $$\partial$$-reducible, $$K$$ is called $$\partial$$-reducing if it admits such a surgery. The 1-bridge conjecture states that if a knot $$K$$ is a $$\partial$$-reducing knot, then either $$K$$ or its dual knot is a 1-bridge knot.
The results of this paper are two-sided. It is first shown that the conjecture is true if there is an essential torus separating the knot from $$\partial M$$. As a consequence, the author gives a complete classification of all satellite $$\partial$$-reducing knots. In the second part the author gives a counterexample to the 1-bridge conjecture and its weakened version, thus settling the problem in general. The third part is a study of the still open problem of whether the conjecture is true for knots in handlebodies. It is shown that if $$K$$ is a 1-tunnel $$\partial$$- reducing knot in a handlebody, then it is a 1-bridge knot. It is also shown that if $$K$$ is a 1-tunnel $$\partial$$-reducing knot, then its dual knot is also a 1-tunnel knot. A corollary of this result is: If a Dehn surgery on a 1-bridge knot in a handlebody yields a handlebody, then the dual knot corresponding to this surgery is also a 1-bridge knot.
Reviewer: Y.-Q.Wu (Austin)

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57N10 Topology of general $$3$$-manifolds (MSC2010)
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